FUNCIONES DE LOS FILTROS DEL LADO DE DC

Hydrologic studies in data-scarce regions use remotely sensed precipitation as meteorolog- ical forcing and satellite-based estimates of evapotranspiration for data assimilation. But remotely sensed P and E datasets exhibit large uncertainty requiring comprehensive val- idation for the area of study. This study addresses this important issue by developing a validation framework that tests for the physical consistency of remotely sensed P and E datasets without the use of concurrent ground-based measurements. A RMSE-based error metric is developed and comprehensively tested to see whether the metric can translate in- dividual biases in P and E datasets onto the Budyko space. Results show that the proposed validation framework is capable of arriving at the same conclusions as traditional validation methodologies regarding the quality of P and E datasets. The application of the developed framework to a data-scarce catchment using publicly available topographic, vegetation and aridity information is also presented. In contrast to previous validation studies that employ complex distributed hydrological models, the use of the single parameter Budyko function highlights the effectiveness of using simple water and energy balance principles in validation of observational data.

Owing to the limitations of the original Budyko formulation, the developed framework can only test whether the combination of P and E datasets can describe the long-term combined water and energy balance of catchments. This implies that the developed RMSEM Dm metric

characterizes the bias in P and E datasets and not the variance. Recent studies have focused on extending the Budyko hypothesis to sub-annual timescales (Greve et al., 2016; Zhang et al., 2008). Therefore, future work involves the extension of the framework to validation of P and E datasets at monthly and daily time-scales, which is crucial for characterizing the variance and also for hydrologic applications such as streamflow forecasting and reservoir operations. In addition, it is assumed that in the long-term, storage in the catchment is negligible. Therefore, care must be taken when applying the framework in catchments which

have long-term storage such as snow, ice or reservoirs and also in small catchments where storage influences water availability. It is seen that the RMSEM Dm metric is more sensitive

to biases in P rather than E. Therefore, care must be taken in interpreting the error metric when the focus of a study is solely on evaluating E datasets which are relatively close to each other. But E datasets can be still be effectively evaluated using the framework if accurate estimates of P are available and the focus of the study is to validate only E datasets

It is to be noted here that the developed framework does not require concurrent ob- servations of precipitation and evapotranspiration for validating remote sensing data. But the application of this framework to a data-scarce region requires reliable estimates of AI, which could be sourced from non-concurrent ground-based measurements, as was done in this study. Although a large sample of catchments, representing a wide range of aridities, have been used in the study, we encourage researchers to validate the robustness of the devel- oped framework in other geographies having different topographic, hydrologic and climatic characteristics.

CHAPTER 3

Calibration of Large Scale Hydrologic Models with

Multiple Fluxes: The Necessity and Value of a Pareto

Optimal Approach

3.1

Introduction

The widespread use of large scale hydrologic and land surface models (LSMs) in snow (Chris- tensen and Lettenmaier, 2007; Li et al., 2017), drought (Leng et al., 2015; Sheffield et al., 2004), and climate change (Middelkoop et al., 2001; Cuo et al., 2013) studies necessitates critical examination of the adopted calibration methodologies. The general approach of cal- ibrating the models with measurements of a single flux, typically streamflow, is considered inadequate for such studies that require other water balance components to be simulated accurately. Rakovec et al. (2016b) evaluate the performance of the mesoscale hydrologic model (mHM) calibrated with streamflow (SF) against observed evapotranspiration (ET), soil moisture (SM), and total water storage (TWS). The study concludes that calibrating hydrologic models with only streamflow may not be sufficient for accurate simulation of other water balance components. Wanders et al. (2014) calibrate the LISFLOOD hydro- logic model with remotely sensed soil moisture datasets. The results of the study show that calibrating the hydrologic model with only soil moisture negatively affects the accuracy of the corresponding streamflow simulation (compared to streamflow-calibrated model results). L´opez L´opez et al. (2017) confirm the findings of the other studies; calibrating a hydrologic model with only ET or SM adversely affects the accuracy of streamflow simulation compared to streamflow-calibrated model results. In Zink et al. (2018), a land surface model calibrated

with land surface temperature leads to higher errors in streamflow simulations compared to a streamflow-calibrated model.

Incorporating multiple fluxes into the calibration process has emerged as a consensus solution to address the adverse effects of single objective calibration. A number of different methods have been employed to calibrate hydrologic and land surface models with multiple fluxes, including stepwise calibration (Sutanudjaja et al., 2013; L´opez L´opez et al., 2017), ensemble Kalman filter (Wanders et al., 2014), and simultaneous calibration by combining objective functions (Rientjes et al., 2013; Rakovec et al., 2016a; Zink et al., 2018). Irrespective of the calibration strategy adopted, all the studies report improvements in the simulation of the added flux or storage component while maintaining the accuracy of the primary variable of interest. The improvements are also consistent across different water balance components incorporated into calibration, including evapotranspiration (Rientjes et al., 2013; L´opez L´opez et al., 2017; Zink et al., 2018), soil moisture (Sutanudjaja et al., 2013; Wanders et al., 2014), and total water storage (Rakovec et al., 2016a). Although the enumerated studies provide evidence in favor of multivariate calibration, we identify shortcomings in these approaches that hinder comprehensive quantification of the value of incorporating additional fluxes.

First, multivariate calibration studies do not define any limits of acceptability or error thresholds to determine whether the model can simultaneously reproduce the incorporated fluxes to a sufficient degree of accuracy. To illustrate the importance of defining limits of acceptability, consider the results of Rakovec et al. (2016a) wherein the addition of TWS estimates into calibration along with streamflow reduce the root mean square error (RMSE) of TWS simulations at negligible cost to the accuracy of streamflow simulation. A closer analysis of the results reveals that despite reduction in the RMSE of TWS, the absolute value of RMSE is still significantly large (Figure 3. in Rakovec et al. (2016a)); whereas the RMSE of standardized anomalies of streamflow has a median of approximately 0.5, the RMSE of TWS is approximately 0.8 (reduced from 0.9 for the SF-calibrated model). Without defining a threshold for acceptable error, it is difficult to assess whether the reported reduction in

TWS error is sufficient evidence to conclude that the incorporation of an additional flux actually improves the realism of the model.

Second, most studies consider the relationship between different fluxes to be comple- mentary but all the results, with the exception of Wanders et al. (2014), point towards a trade-off relationship. By definition, a complementary relationship would mean that in- corporation of additional fluxes improves the accuracy of all the incorporated fluxes. The objective functions of calibration and the calibration methodologies (such as stepwise cali- bration) are constructed to reflect the assumption of a complementary relationship. Even in Wanders et al. (2014) the improvement in SF accuracy when SM is incorporated, compared with a streamflow-calibrated model, is limited to small catchments. In addition, the results of multivariate calibration rarely are compared with results of models calibrated only with the additional flux. Such a comparison would help in quantifying the potential trade-offs in simulating the two fluxes accurately. For example, in Rakovec et al. (2016a) the model is not calibrated with only TWS, which would help understand the trade-off in TWS accuracy required to achieve acceptable SF accuracy. In studies where all the calibration cases are reported, there are significant trade-offs among the different fluxes considered for calibration (Rientjes et al., 2013). Even in Zink et al. (2018), where the objective function is designed to produce a compromise solution between the SF and ET fluxes, there is no discussion of either the magnitude of trade-off in the accuracy of ET or of whether such trade-offs are within acceptable limits.

Third, the trade-off relationship among the simulated fluxes implicit in the results of multivariate calibration studies may be a consequence of deficiencies in model structure and parameterizations (Fenicia et al., 2007; Hogue et al., 2006). However, in the calibration strategies adopted in most studies, including the assumption of a complementary relationship among the fluxes, combining objectives and lack of a definition of error thresholds prevent any meaningful diagnoses of the model. For example, the limitations of the stepwise cal- ibration methodology for identifying deficiencies in model structure and parameterizations is well known (Fenicia et al., 2007). Additionally, most multivariate calibration studies are

deterministic and hence are inappropriate for studying the dierences in optimal parameter sets between univariate and multivariate calibration cases, as they do not address the issue of equifinality (Beven, 1996, 2001). Even in studies that use stochastic methods such as ensemble Kalman filter (Wanders et al., 2014), there is little discussion on how parameters behave between different calibration cases.

In this study, we combine a formal Bayesian calibration approach with the concept of Pareto optimality to address the issues detailed above. We utilize a formal Bayesian calibra- tion approach to define the limits of acceptability or error thresholds in order to distinguish between behavioral and non-behavioral solutions (Beven, 2006; Vrugt et al., 2009b) for each of the incorporated water balance components. Behavioral solutions are model parameter sets that result in errors that are within a defined threshold or limit with respect to a spe- cific simulated response (for example ET or SM). If a trade-off relationship does exist among the incorporated fluxes, as opposed to a complementary relationship, the concept of Pareto optimality would help in understanding the extent to which the accuracy of a particular flux can be improved without affecting, to an unreasonable degree, the accuracy of other fluxes. In addition, Pareto optimal solutions are unbiased by any subjective weights given to any particular flux or storage component over another, unlike simultaneous calibration strategies (Gupta et al., 1998). Hence, we use Pareto optimality-based calibration to create a set of non-dominated solutions that characterize the trade-offs among the incorporated variables. We develop a multivariate calibration framework that combines behavioral solutions from Bayesian calibration and multivariate calibration solutions to address the following research questions: 1) Does incorporation of multiple fluxes into calibration produce parameter dis- tributions that are behavioral with respect to all fluxes considered for calibration? 2) For a given large scale hydrologic model, what is the extent of trade-off, if any, in accurate sim- ulations of multiple fluxes considered for calibration? 3) Can behavioral and multivariate calibration solutions help identify deficiencies in hydrologic model parameterization that lead to trade-offs in the accurate simulations of multiple variables?